Physics • Year 11 • Module 2 • Lesson 7
Gravitational PE and Energy Conservation
Build HSC Band 5–6 extended-response technique on energy analysis, experimental design, and the physics of why mass cancels from the conservation equation.
1. Data + scenario: rollercoaster energy audit at a real theme park (Band 5–6)
8 marks Band 5–6
Scenario. Engineers tested a rollercoaster at a theme park. A 600 kg car was released from rest at the top of a 45 m drop. Sensors recorded the speed at four points along the track. The engineers also placed a second car with mass 1200 kg (twice as heavy) on the same track and recorded its speeds. The data are shown below.
| Position | Height above bottom (m) | Speed — 600 kg car (m/s) | Speed — 1200 kg car (m/s) |
|---|---|---|---|
| A (start) | 45 | 0 | 0 |
| B (quarter-way down) | 33.75 | 14.8 | 14.8 |
| C (bottom of first drop) | 0 | 27.1 | 27.1 |
| D (top of second hill) | 18 | 21.1 | 21.1 |
| E (valley) | 5 | 24.4 | 24.4 |
Illustrative data. Use g = 9.8 m s−2. Speeds are rounded to 3 significant figures.
Q1. Analyse and evaluate the experimental data above. In your response you must:
- Verify algebraically that the speed at position C is consistent with frictionless conservation of energy, showing your working from KE1 + U1 = KE2 + U2.
- Use the data to explain why the 600 kg and 1200 kg cars reach identical speeds at every position, deriving the algebraic reason mass cancels.
- Determine whether mechanical energy is conserved between A and D by comparing calculated ideal speed at D with the data value. Show your reasoning quantitatively.
- Calculate the energy lost to friction between A and D for the 600 kg car, and find the average resistance force if the track distance A–D is 85 m.
- State one limitation of concluding that the track is “frictionless” based solely on the fact that both cars reach the same speed.
2. Experimental design — testing the mass-independence of speed at the bottom of a ramp (Band 5–6)
7 marks Band 5–6
Research question. A student claims: “A heavier ball will reach a higher speed at the bottom of a ramp because it has more gravitational PE at the top.” Design a scientific investigation to test whether the speed of a ball at the bottom of a ramp is independent of mass when dropped from the same height.
Constraints: You have access to a smooth ramp (approximately frictionless), a metre ruler, a digital stopwatch, steel balls of masses 50 g, 100 g, and 200 g, and a motion sensor. Your investigation must be completable in a single 75-minute lesson.
Q2. Design the investigation and present it in the format below.
- State your hypothesis (a testable prediction including independent and dependent variables).
- Identify the independent variable, dependent variable, and at least two controlled variables.
- Describe the procedure in at least four numbered steps, including how you will measure speed at the bottom and how you will control height.
- Explain what result would falsify your hypothesis.
- State two limitations of your design and one way to improve reliability.
Q1 — Sample Band 6 response (8 marks), annotated
Verify speed at C (2 marks):
Frictionless prediction: v = √(2gh) = √(2 × 9.8 × 45) = √882 = 29.7 m/s. The data value at C is 27.1 m/s, which is less than 29.7 m/s [1]. This confirms friction is present — the track is not completely frictionless [1]. Award 1 mark for a correct calculation of 29.7 m/s; 1 mark for comparing with data and concluding friction is present.
Why mass cancels — algebraic derivation (2 marks):
From KE1 + U1 = KE2 + U2: 0 + mgh1 = ½mv² + mgh2. Divide every term by m: gh1 = ½v² + gh2. Mass m has cancelled completely from the equation [1]. Therefore the speed at any point depends only on g, h1, and h2 — not on the mass of the car. This is why the 600 kg and 1200 kg cars reach identical speeds at every position [1].
Mechanical energy conservation between A and D (2 marks):
Ideal frictionless speed at D (starting from A at h = 45 m, D at h = 18 m): vD = √(2g(hA − hD)) = √(2 × 9.8 × (45 − 18)) = √(529.2) = 23.0 m/s. The data value at D is 21.1 m/s < 23.0 m/s [1]. Since the actual speed is less than the frictionless prediction, mechanical energy is NOT conserved between A and D; energy has been lost to friction [1].
Energy lost and friction force (2 marks):
Emech,A = mghA = 600 × 9.8 × 45 = 264 600 J. Emech,D = ½mvD² + mghD = ½ × 600 × 21.1² + 600 × 9.8 × 18 = 133 503 + 105 840 = 239 343 J. Elost = 264 600 − 239 343 = 25 257 J ≈ 25 300 J [1]. Average friction force F = Elost / s = 25 257 / 85 = 297 N ≈ 300 N [1].
Limitation (1 mark): That both cars reach the same speed demonstrates that speed is mass-independent, which is consistent with a frictionless prediction. However, mass-independence also holds when friction is present (mass cancels from the energy equation even with friction, provided the same geometry and surface conditions apply to both cars). Therefore, equal speeds for both masses cannot distinguish a frictionless track from a track with friction. A better test would be to compare the actual speed against the ideal frictionless prediction, as done in part 3 above [1].
Marking criteria summary (8 marks): 1 = correct frictionless speed at C (√882 = 29.7 m/s). 1 = uses data comparison to conclude friction present. 1 = algebraic derivation of mass cancellation from energy equation. 1 = concludes speed is mass-independent from algebra. 1 = correct ideal speed at D (23.0 m/s). 1 = compares with data (21.1 m/s) and concludes mechanical energy not conserved. 1 = correct Elost calculation using before/after Emech. 1 = correct friction force F = Elost/s.
Q2 — Sample Band 6 response (7 marks), annotated
Hypothesis: If speed at the bottom of a ramp is independent of mass (as predicted by v = √(2gh)), then steel balls of 50 g, 100 g, and 200 g released from the same height will reach the same speed at the bottom of the ramp within experimental uncertainty. Independent variable: mass of ball (50 g, 100 g, 200 g). Dependent variable: speed of ball at the bottom of the ramp (measured by motion sensor in m/s). Controlled variables: height of release from the bottom (h, measured with metre ruler to ±1 mm); ramp surface (same ramp, no lubricant added); release from rest (ball held then released, not pushed). [1 — testable hypothesis with IV and DV clearly stated]
Procedure: (1) Set up the ramp on a level bench. Measure and mark a release height of h = 0.50 m above the bottom using a ruler. (2) Place the 50 g ball at the release point (h = 0.50 m) and hold it stationary. Trigger the motion sensor positioned 10 cm from the base of the ramp to record speed as the ball passes. Release the ball from rest; record the speed at the bottom. Repeat three times and calculate the mean speed. (3) Repeat step 2 with the 100 g ball, then the 200 g ball, keeping h = 0.50 m constant. (4) Compare the mean speeds for all three masses. Calculate the predicted frictionless speed v = √(2 × 9.8 × 0.50) = 3.13 m/s and compare with measured values. [1 — four numbered steps including speed measurement method and height control]
Falsification: The hypothesis would be falsified if the 200 g ball consistently and significantly reached a higher speed at the bottom than the 50 g ball under the same height conditions, after accounting for measurement uncertainty. A difference greater than ±0.05 m/s across all three trials would indicate a mass-dependent effect not predicted by v = √(2gh). [1]
Limitations: (1) The ramp is described as “approximately frictionless” but is not truly frictionless: larger, heavier balls may experience slightly greater rolling friction due to greater normal force, making a fair test between masses difficult. [1] (2) The balls roll, not slide, so rotational kinetic energy is also present. The motion sensor records translational speed, not total KE. This means the measured speed will be slightly less than the sliding-only prediction v = √(2gh), and the discrepancy may differ between balls if their moments of inertia differ. [1]
Improvement: Repeat each measurement five times (not three) and report mean ± standard deviation. Use a digital video camera to independently verify the motion sensor readings at the bottom. [1]
Expected result: All three masses reach the same speed (≈ 3.13 m/s for h = 0.50 m, slightly less due to friction and rolling), confirming that speed at the bottom is mass-independent. [1 — explains expected outcome and links it to the energy derivation]
Marking criteria summary (7 marks): 1 = testable hypothesis naming IV and DV; 1 = four steps with speed measurement and height control; 1 = states a specific falsification criterion; 1 = valid limitation 1 (friction / surface); 1 = valid limitation 2 (rolling/rotation or reaction-time); 1 = one improvement to reliability; 1 = links expected result back to mass cancellation in the energy equation.